It's actually the opposite (to a point, I think). Higher mean free path means more time to accelerate, then hit and ionize something else. That's how fluorescent tubes maintain discharge a meter long at measly voltages.

Radiation doesn't help either, as sibling commented - it creates ions making the initial breakdown easier, though it shouldn't make a difference for a continuous flow that involves innumerable ions.

Here ya go:

https://www.youtube.com/watch?v=MIqK45nxcTo

He even explained it with balls and magnets!

I would guess that it could be due to having more ions in air due to higher radiation levels.

Yes, and when the cathode heater burns out, they stop conducting, as they depend on the thermionic emission of electrons (and the absense of gases).